Question

 If α and β are zeros of polynomial 3x² -x+2 , then form a polynomial whose zeroes are 3α and 3β
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Answer 1

Answer:

Step-by-step explanation:

Here, we have  α and β as our zeroes.

What we know is

α + β = -b/a

and, we also know that

α . β = c/a

Now, what we're going to do is apply the same for the polynomial 3x² -x+2

• Where we've a as 3
• Where we've b as -1
• Where we've c as 2

Check in:-

α + β = -(-1)/3

α + β = 1/3

α . β = 2/3

Now, we've the new zeroes as 1/3 and 2/3.

Let us assume that :

λ = 1st zero = 1/3

μ = 2nd zero = 2/3

Now, we'll be having 3λ+3μ as new -b/a and 9μλ as c/a.

 = 3λ + 3μ = 3(λ + μ) = 3 x 1/3 = 1

= 3λ + 3μ = 1.

= 3λ * 3μ

= 9λμ

= 9 * 2/3 = 6  

= x² - (λ + μ) x + aμ

= 1x² - (1)x + 6

= 1x² - x + 6 is the required answer.

Answer 2

⠀⠀⠀⠀⠀⠀⠀⠀⠀✠ Given Polynomial : 3x² - x + 2

⠀⠀⠀⠀By Comparing , the Given Polynomial with Standard form of Quadratic Equation ( ax² + bx + c ) , we get —

• a = 3 ,
• b = – 1 &
• c = 2 .

As , We know that ,

• ( α + β ) = –(b)/a &
• (αβ) = c/a

⠀⠀⠀⠀⠀⠀⠀★ By Substituting The Known Values :

⇢ ( α + β ) = -(b)/a

⇢( α + β ) = – (-1)/3

⇢( α + β ) = ⅓ ★

⠀⠀⠀⠀&

⇢ ( α β ) = c/a

⇢( α + β ) = 2/3

⇢( α + β ) = ⅔ ★

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

☯︎ Finding Quadratic Polynomial whose zeroes are 3α and 3β .

As , We know that ,

⠀⠀⠀⠀⠀⠀⠀ Quadratic Equation = x² - (α+β)x + (αβ)

⇢ x² - (α+β)x + (αβ)

⇢ x² - ( 3α + 3β )x + ( 3α × 3β )

⇢ x² - { 3 ( α + β ) }x + ( 9αβ )

⠀⠀⠀⠀⠀⠀⠀★ By Substituting The Known Values :

• ( α + β ) = ⅓ &
• ( α β ) = ⅔

⇢ x² - { 3 ( α + β ) }x + ( 9αβ )

⇢ x² - { 3 ( ⅓ ) }x + { 9 ( ⅔ ) }

⇢ x² - ( 1 )x + (6)

⇢ x² - x + 6 ★

∴ Hence , The Quadratic Polynomial is x² - x + 6 .